# The CNOT Quantum Logic Gate Using q-Deformed Oscillators

###### Abstract

It is shown that the two qubit CNOT (controlled NOT) gate can also be realised using q-deformed angular momentum states constructed via the Jordan-Schwinger mechanism.Thus all the three gates necessary for universality i.e.Hadamard, Phase Shift and the two qubit CNOT gate are realisable with q-deformed oscillators.

Keywords: universality of quantum logic gates ; q-deformed oscillators ; quantum computation

PACS: 03.67.Lx ; 02.20.Uw

1. Introduction

Recently it has been shown that the single qubit quantum logic gates, viz. ,the Hadamard and Phase Shift gates can also be realised with two q-deformed oscillators where is the deformation parameter of a quantum Lie algebra. q-Deformed oscillators here mean that the Lie algebra satisfied by creation () and annihilation () operators of a bosonic oscillator, viz. is modified into where is the number operator and . Using such deformed oscillators an alternative formalism for quantum computation can be set up. The advantage of this over the conventional formalism (which is obtained for ) is the presence of an arbitrary function which may be exploited for experimental purposes.

However, the formalism will be more meaningful if the realisation with q-deformed qubits is possible for all the gates required for universality. A set of gates is said to be universal for quantum computation if any unitary operation may be approximated to arbitrary accuracy by a quantum circuit involving those gates. In the case of standard quantum computation the Hadamard, Phase Shift and the CNOT (controlled NOT) gates constitute such a set. In this paper I show that the 2-qubit controlled-NOT gate can also be realised with q-deformed qubits. Thus all the three gates, i.e. Hadamard, Phase Shift and CNOT gates can now be obtained with q-deformed qubits.

The motivation for considering q-deformed oscillators in quantum computation comes from the fact that deformed oscillators have been successfully used as a tool to understand deviations from an ideal theoretical or experimental scenario for past many years. Bonatsos and Daskoloyamis were among the firsts to show that the vibration spectra of diatomic molecules gave better fits using deformed oscillators. Parisi studied a d-dimensional array of Josephson junctions in a magnetic field and computed the thermodynamic properties in the high temperature region for . Evaluation of the high temperature expansion coefficients were done by mapping onto the computation of some matrix elements for the q-deformed harmonic oscillator. Raychev et al calculated the deviations from the nuclear shell model using the q-deformed three dimensional harmonic oscillator.Bonatsos, Lewis,Raychev and Terziev demonstrated that the three dimensional q-deformed harmonic oscillator correctly predicts the first supershell closure in alkali clusters without introducing additional parmeters.McEWan and Freer showed that the nuclear orbitals of certain nuclei were commensurate with the energy level scheme of the deformed harmonic oscillator and the Nilsson model. For these reasons it is meaningful to study whether deformed oscillators can be used in the formalism of quantum computation.

A brief review of relevant facts is given in section 2. In Section 3 the NOT gate is realised with q-deformed oscillators. Section 4 gives the realisation of the two qubit CNOT gate in terms of q-deformed oscillators. In Section 5 the states are discused and Section 6 is the conclusion where a brief elaboration of the possibility of quantm error correction using deformed oscillators is given.

2.Brief Review

Quantum logic gates are basically unitary operators . Three gates,the single qubit Hadamard and Phase Shift gates and the 2-qubit CNOT gate, are sufficient to construct any unitary operation on a single qubit .This is the universality referred to above. These gates are constructed using the ”spin up” and ”spin down” states of angular momentum i.e., the two possible states of a qubit are usually represented by ”spin up” and ”spin down” states. This is the Jordan-Schwinger construction using two independent harmonic oscillators. A similar construction of q-deformed angular momentum states can be done using two q-deformed oscillators. In Ref.1 it was shown that the Hadamard and Phase Shift gates can also be realised with q-deformed qubits. To achieve this ,the technique of harmonic oscillator realisation of q-oscillators was used. This allows one to set up an alternate quantum computation formalism.

q-Oscillators are described by deformed creation and annihilation operators, , ,respectively. For ordinary oscillators these are and . , and the deformed oscillators satisfy the following relations :

where the q-number becomes the ordinary number when (i.e.). is the number operator for the q-deformed oscillators and is any function of . The eigenvalue of the number operator denotes the number of bosonic particles. We confine to real . and are related as

is the number operator for usual oscillators with eigenvalue ; and , are arbitrary functions of only with for .

If all these arbitrary functions are unity, then , i.e. the number operator of the deformed oscillator becomes identical to the number operator of the standard harmonic oscillator. But oscillator states are usually expressed in occupation number basis. So if the number operators are identical, there is no way of differentiating between a deformed oscillator state and a standard oscillator state. So nothing is gained and we are still in the realm of standard quantum computation. But equations are general if the are not all equal to unity. Let . Now (equation ). This will be reflected in the Jordan-Schwinger construction of angular momentum states and the states in the two cases will be distinguishable through the function . Further details are in Ref.[1].

We now express a single qubit state in terms of two harmonic oscilator states using the Jordan-Schwinger construction.

(a) States are defined by the total angular momentum and -component of angular momentum i.e. . A particular state is created by acting the creation operators on the vacuum or ground state which is a direct product state of the individual oscillator ground states :

is the ground state (), while are the oscillator ground states. where are the eigenvalues of the number operators of the two oscillators.

(b) A qubit can be either ”up” or ”down” i.e. there are two possible configurations. So the oscillator number operators can take the following sets of values only : (, ”up” state) and (, ”down” state). Hence . As for both qubit states, suppress for simplicity of notation :

Equivalently, in terms of these are

(c)The basis states are ( state and state )

Thephysical meaning of the notation is as follows. The (spin ”up”) state is constructed out of two oscillator states where the first oscillator state has occupation number while the other has occupation number . The ( spin ”down”) state corresponds to the first oscillator having occupation number and the second oscillator having occupation number . So any qubit state is :

represents one of the two possible qubit states while represents oscillator ground state i.e. occupation number ; represents an oscillator state with occupation number etc.

Now consider the Hadamard transformation. For a standard qubit, the Hadamard gate acts as follows. When one of the basis states is given as an input, the output is a superposition of the two basis states,i.e. and . So the Hadamard transformation on a single qubit state () can be represented as (modulo )

i.e.

Following the discussion preceding equation , the general q-deformed state is ;

So the Hadamard transformation for q-deformed state is

This simplifies to:

where

is always or so as to correspond to the qubit. It is simple to check that the q-number is equal to the ordinary number and simlarly the q-number equals odinary number . Hence the q-numbers are always the usual numbers . Same restrictions also apply to usual (i.e.undeformed) oscillators. So we restrict the hatted number operators, and , by where is the identity operator.

The Phase Shift transformation of qubit states is : i.e and . which in our notation is is the phase shift. Now one can proceed as described in the previous sections. Details are in Ref.[1]. There it was shown that both the Hadamard and Phase Shift transformations can be realised with q-deformed qubits. Below it is shown that the same is possible for both the NOT gate and the CNOT gate. where

3.The NOT gate

The NOT gate essentially flips a qubit, i.e. and . It acts on a qubit as : where . In our notation this is . In terms of q-deformed oscillator states this becomes . For q-deformed states this means

i.e. the exponents of the two creation operators are interchanged. Rewritten in terms of the functions this looks like

where one has used the fact that and followed the arguments after equation of Ref.1, relabelled as etc. Using one gets

With respect to the states , the above expression would be indistinguishable from the usual ”NOT” transformation if

which simplifies to

for both and .

Written in terms of its eigenvalues means

This has the solution (say) for both and . Thus the NOT gate is realisable with deformed qubits.Moreover, the conditions for realisation is the same (i.e. ) as for the Hadamard and Phase Shift gates.

4. The CNOT gate

The Controlled-NOT gate is a two-qubit operator where the first qubit is the control and the second qubit the target. The action of the CNOT gate is defined by the following transformations:

where etc. The first line of the transformation signifies that when the control qubit is in the -state , the target qubit does not change after the action of the CNOT gate. The second line means that if the control qubit is in the -state, target qubit changes value after the action of the CNOT gate. This may be written as (modulo constants) as i.e.

Let the oscillators corresponding to the qubit be denoted by and those corresponding to the qubit be .Then in terms of oscillator states the CNOT transformation reads:

where and are the eigenvalues of the number operators corresponding to the respective oscillators with and , denote the ground states corresponding to oscillators and respectively. Writing,

the equation for the CNOT transformation looks like:

In all subsequent discussions we shall use this form . However, for completeness, we note that the CNOT transformation can also be written in the alternative notation as

i.e.

For deformed qubits the CNOT transformation will be

or in terms of deformed oscillators :

As in Ref.1, the harmonic oscillator realisations for the operators and respectively are written in terms of the two functions and as:

and are the respective number operators with eigenvalues and and

Using these expressions in (and relabeling as and as etc.) and suppressing the dependence in and to avoid cumbersome notation one gets

Denoting

the equation becomes

Multiplying both sides of by gives

Note that with respect to the states , will be indistinguishable from the usual CNOT transformation if (the identity operator) i.e. or

Equation is an identity for both and . Therefore the condition is always realisable in the domain of . So the two qubit CNOT gate can be realised with q-deformed oscillators. Hence all the gates required for universality can also be realised with q-deformed oscillators. This implies that any quantum logic gate can be realised with q-deformed oscillators. Thus quantum computation has an alternative formalism.

5. The possible states

There are two possibilities as regards the arbitrary functions .

Case:1 All of them are unity and hence and similarly . So just relates the opertors with .A similar argument holds for the operators and . Also from we then have and . This means that at the occupation number level the deformed states cannot be distinguished from the undeformed states and we are in the realm of standard quantum computation. We denote eigenvalues of the number operators for deformed oscillators in Case I by ( still correspond to undeformed oscillators); the states in Case I by .Then relabel by etc. we have for Case I

where and . Note that all states have .The possible states are: